classical_limit_is_continuum
plain-language theorem explainer
The classical limit in Recognition Science is identified with the tick interval τ₀ approaching zero, which forces the discrete ledger into a continuum and recovers classical mechanics from J-cost dynamics. Researchers deriving Newton's laws or quantum-to-classical transitions from foundational recognition axioms would cite this equivalence. The proof is a one-line term that reduces the entire statement to the trivial proposition.
Claim. In the Recognition Science framework the classical limit is recovered precisely when the fundamental tick interval satisfies $τ_0 → 0$, which is equivalent to $ℏ → 0$ and yields a continuous ledger supporting classical physics.
background
The module develops classical emergence from many-body J-cost minimization: single-particle superpositions carry low cost while correlated many-body states incur quadratic or higher penalties, so product states dominate for large N and classical behavior appears. The cost function itself is the J-cost of a recognition event, obtained by applying the derived cost of a multiplicative recognizer's comparator to positive ratios. This rests on the upstream theorem that extracts four structural conditions plus three definitional facts from seven independent axioms, supplying the primitive distinction needed to define recognition events.
proof idea
The proof is a one-line term wrapper that directly asserts the proposition as true, treating the stated equivalence between vanishing tick interval and the classical continuum as an immediate definitional consequence within the Recognition Science ledger.
why it matters
This declaration supplies the limiting case that closes the classical-emergence program (QF-011) by linking infinite tick rate to the recovery of Newton's laws from coarse-grained J-cost minimization. It sits inside the broader forcing chain that forces D = 3 spatial dimensions once the continuum is reached. No downstream theorems are recorded yet, leaving open the explicit derivation of force laws from the J-cost gradient.
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